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On the integral of the squared periodogram

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  • Deo, Rohit S.
  • Chen, Willa W.
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    Abstract

    Let X1,X2,...,Xn be a sample from a stationary Gaussian time series and let I(·) be the sample periodogram. Some researchers have either proved heuristically or claimed that under general conditions, the asymptotic behaviour of is equivalent to that of the discrete version of the integral given by , where [lambda]i are the Fourier frequencies and [phi] and [eta] are suitable possibly non-linear functions. In this paper, we prove that this asymptotic equivalence is not true when [phi] is a non-linear function. We derive the exact finite sample variance of when {Xt} is Gaussian white noise and show that it is asymptotically different from that of . The asymptotic distribution of is also obtained in this case. The result is then extended to obtain the limiting distribution of when {Xt}is a stationary Gaussian series with spectral density f(·). From these results, the limiting distribution of the integral version of a goodness-of-fit statistic proposed in the literature is obtained.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 85 (2000)
    Issue (Month): 1 (January)
    Pages: 159-176

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    Handle: RePEc:eee:spapps:v:85:y:2000:i:1:p:159-176

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    Related research

    Keywords: Periodogram Non-linear functions Box-Pierce statistic Goodness-of-fit;

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    Cited by:
    1. McElroy, Tucker & Holan, Scott, 2009. "A local spectral approach for assessing time series model misspecification," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 604-621, April.
    2. Carlos Velasco & Ignacio N. Lobato, 2004. "A simple and general test for white noise," Econometric Society 2004 Latin American Meetings 112, Econometric Society.
    3. Rea, William & Oxley, Les & Reale, Marco & Brown, Jennifer, 2013. "Not all estimators are born equal: The empirical properties of some estimators of long memory," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 93(C), pages 29-42.
    4. Soulier, Philippe, 2001. "Moment bounds and central limit theorem for functions of Gaussian vectors," Statistics & Probability Letters, Elsevier, vol. 54(2), pages 193-203, September.
    5. Jennifer Brown & Les Oxley & William Rea & Marco Reale, 2008. "The Empirical Properties of Some Popular Estimators of Long Memory Processes," Working Papers in Economics 08/13, University of Canterbury, Department of Economics and Finance.
    6. Faÿ, Gilles, 2010. "Moment bounds for non-linear functionals of the periodogram," Stochastic Processes and their Applications, Elsevier, vol. 120(6), pages 983-1009, June.

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